Determinantal random point fields

By Alexander Soshnikov. 2000

753 citations recorded by the source corpus.

This paper contains an exposition of both recent and rather old results on determinantal random point fields. We begin with some general theorems including proofs of necessary and sufficient conditions for the existence of a determinantal random point field with Hermitian kernel and of a criterion for weak convergence of its distribution. In the second section we proceed with examples of determinantal random fields in quantum mechanics, statistical mechanics, random matrix theory, probability theory, representation theory, and ergodic theory. In connection with the theory of renewal processes, we characterize all Hermitian determinantal random point fields on and with independent identically distributed spacings. In the third section we study translation-invariant determinantal random point fields and prove the mixing property for arbitrary multiplicity and the absolute continuity of the spectra. In the last section we discuss proofs of the central limit theorem for the number of particles in a growing box and of the functional central limit theorem for the empirical distribution function of spacings.

Subfields: Geometry and Topology, Mathematical Physics, Statistics and Probability

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Citations in the corpus

Determinantal Point Processes for Machine Learning (2012, background): “Many good mathematical surveys are now available [Borodin, 2009, Hough et al., 2006, Shirai and Takahashi, 2003a,b, Lyons, 2003, Soshnikov, 2000, Tao, 2009].”
Random matrix theory in statistics: A review (2014, background): “, 2009; Deift, 2000; Deift and Gioev, 2009; Forrester, 2010; Guionnet, 2009; Mehta, 2004; Tao, 2012), and the survey articles (Diaconis, 2003; Soshnikov, 2000).”
The Tracy–Widom limit for the largest eigenvalues of singular complex Wishart matrices (2008, background): “I am thankful to an anonymous referee for valuable suggestions on how to improve the presentation of my results.”
The Random Matrix Theory of the Classical Compact Groups (2019, background): “[97] Alexander Soshnikov.”
Zeros of Random Analytic Functions (2006, background): “The extensive survey of Soshnikov (37) gives many more examples and details.”
Random complex zeroes, I. Asymptotic normality (2002, background): “We mention a recent discovery of Peres and Virág [21] that in the special case L = 1 of the hyperbolic model, the point process is a determinantal one [26].”
Hyperuniform states of matter (2018, background): “Soshnikov [227] presented a review of this subject and discussed applications to random matrix theory, statistical mechanics, quantum mechanics, and representation theory.”
Determinantal probability measures (2002, background): “We remark that when A∪B is finite, the subspace HA,B is closed, since the sum of two closed subspaces, one of finite dimension, is always closed (Halmos (1982), Problem 13) and since the intersection of closed subspaces is always closed.”
Determinantal Processes and Independence (2005, background): “, Burton and Pemantle [4], Soshnikov [27], Lyons [20], Lyons and Steif [21], Shirai and Takahashi [25], Johansson [14], Borodin, Okounkov and Olshanski [3], and Diaconis [8].”
Monte Carlo with determinantal point processes (2016, methodology): “For further information on determinantal point processes, we refer the reader to [Macchi, 1975, Hough et al., 2006, Johansson, 2006, Soshnikov, 2000, Lyons, 2003, Kulesza and Taskar, 2012, Lavancier et al., 2015].”
The Arctic circle boundary and the airy process (2003, methodology): “The paper is a continuation of the approach used in [14] and [15], where certain point processes with determinantal correlation functions [ 24 ] and the Krawtchouk ensemble, were used.”
Determinantal Point Processes (2016, background): “5 (Theorem 4 in [34], see also [18]) Let K be an operator satisfying (H1) and η the determinantal point process with kernel K.”
Random matrix theory (2019, background): “For more, see Soshnikov [21] or Peres [13].”
Scale Invariance of the PNG Droplet and the Airy Process (2001, background): “In the probabilistic literature point processes of this structure are known as determinantal [12].”